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Research Papers

Snap Transitions of Pressurized Graphene Blisters

[+] Author and Article Information
Peng Wang, Kenneth M. Liechti, Rui Huang

Department of Aerospace Engineering and
Engineering Mechanics,
University of Texas,
Austin, TX 78712

Contributed by the Applied Mechanics Division of ASME for publication in the JOURNAL OF APPLIED MECHANICS. Manuscript received February 28, 2016; final manuscript received April 3, 2016; published online April 20, 2016. Editor: Yonggang Huang.

J. Appl. Mech 83(7), 071002 (Apr 20, 2016) (14 pages) Paper No: JAM-16-1115; doi: 10.1115/1.4033305 History: Received February 28, 2016; Revised April 03, 2016

Blister tests are commonly used to determine the mechanical and interfacial properties of thin film materials with recent applications for graphene. This paper presents a numerical study on snap transitions of pressurized graphene blisters. A continuum model is adopted combining a nonlinear plate theory for monolayer graphene with a nonlinear traction–separation relation for van der Waals interactions. Three types of blister configurations are considered. For graphene bubble blisters, snap-through and snap-back transitions between pancake-like and dome-like shapes are predicted under pressure-controlled conditions. For center-island graphene blisters, snap transitions between donut-like and dome-like shapes are predicted under both pressure and volume control. Finally, for the center-hole graphene blisters, growth is stable under volume or N-control but unstable under pressure control. With a finite hole depth, the growth may start with a snap transition under N-control if the hole is relatively deep. The numerical results provide a systematic understanding on the mechanics of graphene blisters, consistent with previously reported experiments. Of particular interest is the relationship between the van der Waals interactions and measurable quantities in corresponding blister tests, with which both the adhesion energy of graphene and the equilibrium separation for the van der Waals interactions may be determined. In comparison with approximate solutions based on membrane analyses, the numerical method offers more accurate solutions that may be used in conjunction with experiments for quantitative characterization of the interfacial properties of graphene and other two-dimensional (2D) membrane materials.

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Figures

Grahic Jump Location
Fig. 1

Three types of graphene blisters: (a) a circular bubble blister with radius a and height h; (b) a center-island blister; and (c) a center-hole blister

Grahic Jump Location
Fig. 2

Normalized traction–separation relation for van der Waals interactions between graphene and its substrate

Grahic Jump Location
Fig. 3

Pressure versus height for a nanoscale graphene bubble blister (a = 10 nm), showing the snap transitions from A to B and from C to D. The dotted line is the unstable branch from A to C. The linear solution and the approximate membrane solution are shown as dashed lines for comparison.

Grahic Jump Location
Fig. 4

Snap transitions of a nanoscale graphene bubble blister (a = 10 nm): (a) Snap-through of the deflection profile from A to B at p= 243 MPa; (b) distributions of the vander Waals force at A and B; (c) snap-back of the deflection profile from C to D at p=142 MPa; and (d) distributions of the van der Waals force at C and D. The points A–D refer to those marked in Fig. 3.

Grahic Jump Location
Fig. 5

(a) Pressure versus height for a microscale graphene bubble blister (a = 1.5 μm). (b) Pressure–height in a log–log plot. (c) Pressure versus volume for the microbubble blister.

Grahic Jump Location
Fig. 6

Evolution of deflection profile for a microscale graphene bubble blister (a = 1.5 μm): (a) and (b) for increasing pressure along branch I (stable), with (b) showing the deflection near the edge; (c)–(e) for decreasing pressure along the unstable branch, with (c) showing the deflection near the center and (d) showing the deflection near the edge; (f) for increasing pressure along branch II (stable)

Grahic Jump Location
Fig. 7

Phase diagrams for graphene bubble blisters: (a) pressure versus radius and (b) height versus radius

Grahic Jump Location
Fig. 8

(a) Central height versus pressure and (b) volume versus pressure for a center-island graphene blister (a = 1.5 μm and b = 0.25 μm)

Grahic Jump Location
Fig. 9

Deflection profiles of a center-island graphene blister (a = 1.5 μm and b = 0.25 μm): (a) donut-like profiles (stable branch I), (b) and (c) delamination and popping (unstable branch), and (d) dome-like profiles (stable branch II)

Grahic Jump Location
Fig. 10

Snap-back transition for a center-island graphene blister (a = 1.5 μm and b = 0.25 μm): (a) critical pressure and (b) pull-in distance. The analytical solutions from Liu et al. [9] are shown for comparison.

Grahic Jump Location
Fig. 11

Critical pressure for snap-through transition of a center-island graphene blister (a = 1.5 μm and b = 0.25 μm), as a function of the adhesion energy Γ. The predictions by the membrane analysis in the Appendix and the analytical model in Boddeti et al. [10] are shown in comparison with the numerical results (symbols).

Grahic Jump Location
Fig. 12

(a) Pressure–volume curve for a center-hole graphene blister (a = 1.5 μm and b = 0.25 μm). The dashed lines correspond to the ideal gas law, pV=NkT, with different values of N as indicated (T = 300 K). (b) Deflection profiles for increasing number of gas molecules. The dashed lines correspond to the critical points B and C in (a).

Grahic Jump Location
Fig. 13

(a) Central height, (b) pressure, and (c) the change of radius for a center-hole graphene blister (a = 1.5 μm and b = 0.25 μm). Dashed lines show the predictions by the approximate membrane analysis.

Grahic Jump Location
Fig. 14

Calculated delamination resistance curves for a center-hole graphene blister (a = 1.5 μm and b = 0.25 μm) using two different formulas based on the approximate membrane analysis

Grahic Jump Location
Fig. 15

Pressure–volume curves, (a) for unstable growth of a center-hole graphene blister (d = 1.0 μm) and (b) for stable growth with d = 0.01 μm, both under N-control. The dashed lines correspond to the ideal gas law with different values of N as indicated (T = 300 K).

Grahic Jump Location
Fig. 16

Shape functions for graphene bubble blisters

Grahic Jump Location
Fig. 17

Comparison of the approximate two-state solution by energy minimization with the numerical solution for the pressure–height curve of a graphene bubble blister (a = 10 nm)

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